Democrats have grown increasingly concerned that oversight of the special counsel’s Russia probe will be wrested away from Deputy Attorney General Rod Rosenstein — either through recusal or dismissal.

President Trump appeared to direct ire over the investigation at the deputy attorney general on Twitter last week, meanwhile Mr. Rosenstein has reportedly privately discussed the possibility that he may need to recuse himself from the matter.

“I am being investigated for firing the FBI Director by the man who told me to fire the FBI Director! Witch Hunt,” Mr. Trump tweeted Friday in an apparent reference to Mr. Rosenstein, who wrote the memo that recommended James B. Comey’s firing.

That left Sen. Dianne Feinstein, the ranking Democrat on the Judiciary Committee, worried about the president’s intentions.

“The message the president is sending through his tweets is that he believes the rule of law doesn’t apply to him and that anyone who thinks otherwise will be fired,” Mrs. Feinstein said. “If the president thinks he can fire Deputy Attorney General Rosenstein and replace him with someone who will shut down the investigation, he’s in for a rude awakening. Even his staunchest supporters will balk at such a blatant effort to subvert the law.”

But Mr. Rosenstein, who made the decision last month to appoint Robert Mueller as the special counsel investigating Russia’s meddling in the 2016 presidential election, could potentially recuse himself from oversight of the matter.

Citing sources within the Justice Department, ABC News reported Friday that Mr. Rosenstein raised the possibility of his own recusal with Associate Attorney General Rachel Brand. As the DOJ’s third-in-command, Ms. Brand would be the next in line to assume oversight of the special counsel.

The Justice Department said Friday that Mr. Rosenstein has not recused himself from the probe.

“As the deputy attorney general has said numerous times, if there comes a point when he needs to recuse, he will. However, nothing has changed,” said DOJ spokesman Ian Prior.

Discussion about a possible recusal, however, could indicate a widening of the scope of the investigation.

“I’ve talked with Director Mueller about this,” Mr. Rosenstein told The Associated Press earlier this month. “He’s going to make the appropriate decisions, and if anything that I did winds up being relevant to his investigation then, as Director Mueller and I discussed, if there’s a need from me to recuse, I will.”

Reports last week indicated the probe has now stretched to include inquiries into the business dealings of Mr. Trump’s son-in-law and adviser Jared Kushner, as well as whether Mr. Trump tried to obstruct the Russia investigation.

Attorney General Jeff Sessions recused himself from the Russia investigation in March, citing his own ties to President Trump’s campaign.

The Democratic National Committee raised concern over the passage of the investigation to another Trump appointee if Mr. Rosenstein were to recuse.

“After Trump’s repeated attempts to enlist top law enforcement officials to interfere in the investigation, there is no reason to believe that anyone within the administration could be effectively insulated from his influence,” DNC spokesman Daniel Wessel said.

Instead, the DNC said Mr. Mueller should be granted total independence and full control over the probe.

The president’s Twitter comments came the morning after Mr. Rosenstein issued a strange public statement that condemned news stories attributed to anonymous sources.

“Americans should exercise caution before accepting as true any stories attributed to anonymous ‘officials,’ particularly when they do not identify the country — let alone the branch or agency of government — with which the alleged sources supposedly are affiliated,” Mr. Rosenstein said. “Americans should be skeptical about anonymous allegations. The Department of Justice has a long-established policy to neither confirm nor deny such allegations.”

Justice Department officials did not provide further context for the statement, but it came within hours after The Washington Post reported that the special counsel’s investigation had expanded to include questions about Mr. Kushner’s business dealings.